In standard VAE models, the generative likelihood of the observed data is modelled as normally distributed. However in real-life applications, such as in healthcare, the data is typically heterogeneous.
In order to effectively capture the different data characteristics, the generative model should be able to handle different likelihood distributions. Nazabal et al. [1]} suggested a VAE model (HI-VAE) that can handle different likelihood parameters and, at the same time, capture the statistical dependencies among the differently distributed data dimensions. The generative model for the \(n\)th heterogeneous data point is
\(p_{\omega }(\mathbf {y}_n, \mathbf {z}_n) &= p_{\theta }(\mathbf {z}_n) \prod _{d=1}^D p_{\psi }(y_{nd} \mid \mathbf {\gamma }_{nd}) \\\mathbf {\gamma }_{nd} &= \mathbf {h}_d(\mathbf {a}_{nd}) \quad \mathbf {A}_n = [\mathbf {a}_{n1},...,\mathbf {a}_{nD}] = \mathbf {g}(\mathbf {z}_n), \nonumber \) 
